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Proceedings of the American Mathematical Society

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Distribution of alternation points in uniform polynomial approximation

Author: G. G. Lorentz
Journal: Proc. Amer. Math. Soc. 92 (1984), 401-403
MSC: Primary 41A50; Secondary 41A10
MathSciNet review: 759662
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Abstract: For a continuous function $ f$ on $ \left[ {0,1} \right]$, we discuss the points where the polynomial $ {P_n}\left( x \right)$ of best uniform approximation deviates most from $ f\left( x \right)$, and the signs of the difference $ f\left( x \right) - {P_n}\left( x \right)$ alternate. We show that these points can be very irregularly distributed in $ \left[ {0,1} \right]$, even if $ f$ is entire.

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  • [1] S. N. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, Gauthier-Villars, Paris, 1926.
  • [2] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • [3] H. Fiedler and W. B. Jurkat, Best 𝐿¹-approximation by polynomials, J. Approx. Theory 37 (1983), no. 3, 269–292. MR 693014, 10.1016/0021-9045(83)90053-9
  • [4] M. Ĭ. Kadec′, On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials, Uspehi Mat. Nauk 15 (1960), no. 1 (91), 199–202 (Russian). MR 0113079
  • [5] J. H. B. Kemperman and G. G. Lorentz, Bounds for polynomials with applications, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 1, 13–26. MR 528214
  • [6] E. B. Saff and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie, Band 4 (Meeting, Math. Forschungsinst., Oberwolfach, 1977) Internat. Schriftenreihe Numer. Math., vol. 42, Birkhäuser, Basel-Boston, Mass., 1978, pp. 281–298. MR 527107

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Article copyright: © Copyright 1984 American Mathematical Society